Multi-stage axial-flow compressor



Sept. 20, 1955 G. L. WlLDE 2,718,349

MULTI-STAGE AXIAL-FLOW COMPRESSOR Filed June 25, 1951 6 Sheets-Sheet 1 V411 QH 31? flgjh VWF W 2L 0L 1% lAll/ENTIJR e. L. WILDE A TTVS.

Sept. 20, 1955 e. WlLDE MULTI-STAGE AXIAL-FLOW COMPRESSOR 6 Sheets-Sheet 2 Filed June 25 1951 T 211; I uwmm/e G.L. WILDE ATTKS'.

Sept. 20, 1955 G. L. WILDE MULTI-STAGE AXIAL-FLOW COMPRESSOR 6 Sheets-Sheet 6 Filed June 25, 1951 nvvzurox G. L. WILDE B 21am) Z747TTYS.

Sept. 20, 1955 G. WILDE MULTI-STAGE AXIAL-FLOW COMPRESSOR Filed June 25, 1951 M/VE/VTDR W ILDE 3 MM +77/ 6 Sheets-Sheet 4 p 0, 1955 G. L. WlLDE 2,718,349

MULTI-STAGE AXIAL-F LOW COMPRESSOR Filed June 25, 1951 6 Sheets-Sheet 5 I lNl/EIVTDR G. L. WILDE ATTVS.

Sept. 20, 1955 wlLDE 2,718,349

MULTI-STAGE AXIAL-FLOW COMPRESSOR Filed June 25, 1951 6 Sheets-Sheet 6 INI/ENTOR ,G. L. W/L DE By -f ATTYS.

United States Patent- O MULTI-STAGE AXIAL-FLOW COMPRESSOR Geoffrey Light Wilde, Coxbench, England, assignor to Rolls-Royce Limited, Derby, England, a British com- Application June 25, 1951, Serial No. 233,373 Claims priority, application Great Britain June 28, 1950 17 Claims. (Cl. 230114) This invention relates to multi-stage axial-flow compressors and has for its objects to provide new and improved means whereby the efliciency of the compressor may be maintained over a wide speed range of operation.

'It is usual to design such a compressor to run at the particular speed which corresponds with the normal operating condition of the compressor. This speed is normally a high proportion of the maximum operating speed of the compressor. The characteristics of the blades of the compressor including the incidence, profile and pitch of the blades are therefore chosen to give the optimum efliciency at the speed at which the compressor is designed to operate. At speeds below the design speed, when the pressure rise per stage is less than the designed pressure rise, there will tend to be an accelerating flow from entry to exit of the compressor due to the design overall density ratio not being achieved. This acceleration appears as a reduction of axial velocity at the entry of the compressor and an increase at the exit.

It has been found that the reduction of axial velocity at the entry to the compressor results in the rotor blades of the initial stages of the compressor, and also in some cases the stator blades of the initial stages, operating at a much higher angle of incidence than in the design condition. When the speed of the compressor is reduced sufiiciently this high angle of incidence results in the blades stalling, the efficiency of the compressor then being seriously reduced. It has been proposed to provide variable pitch stator blades to increase the angle of swirl of the gases leaving the stator blades in order to improve the angle of incidence of the flow of air on to the succeeding rotor blades; it has also been proposed that air should be bled off from a later stage of the compressor, whereby the axial velocity of the air passing through the earlier stages of the compressor is increased.

It has been found that the angle of incidence of the air relative to the blades of the compressor is most ad versely affected over that portion of the blades nearest the tip i. e. over substantially one-third of the spanwise length of the blade toward the outer radius thereof.

An object of the present invention is toimprove the flow conditions in the compressor, particularly over the radially outer portion of the compressor blades.

According to one aspect of this invention therefore, a method of operating a multistage axial-flow compressor comprises the injection of a supply of working fluid into the initial stages of the compressor, in addition to the normal supply of working fluid drawn in through the intake, when the compressor is operating below a selected speed which is less than the design speed.

According to a feature of the invention the supply of additional working fluid is injected into the compressor 2,718,349 Patented Sept. 20, 1955 the outer third of the spanwise length of the blades of the initial stages of the compressor.

The invention in another aspect comprises a multistage aXial-flow compressor comprising means to inject a supply of additional working fluid into the initial stages of the compressor in addition to the normal supply of working fluid drawn in through the compressor intake when the compressor is operating below a selected speed which is less than the design speed.

Preferably the said means is arranged to inject the additional working fluid to a greater extent over the outer portions of the blades than over the radially inner portions of the blades and in one preferred form of the invention the additional working fluid is injected substantially wholly over the outer portions of the blades of the initial stages of the compressor.

According to another aspect of the invention the compressor may comprise means to control the injection of the additional working fluid into the initial stages of the compressor automatically in accordance with an operating variable of the compressor which varies with the speed of the compressor rotor; the control means may include a control device sensitive to the operating variable (such as compressor rotational speed, corrected rotational speed, compressor delivery pressure or compressor compression ratio) and arranged to prevent the injection of the additional working fluid when said operating variable passes through a selected value on increase of speed of the compressor rotor, and to allow the injection of the additional working fluid when the value of the operating variable passes through said selected value on decrease of speed of the compressor rotor. I

According to another feature of the invention the means to supply additional working fluid may be arranged to inject the additional working fluid into the intake of the compressor prior to the first row of rotor blades and preferably prior to the inlet guide vanes.

According to another feature of the invention the means to supply additional working fluid for injection into the compressor may comprise means for-bleeding off air at a later stage in the compressor. 7

According to another feature of the invention, a multistage axial-flow compressor having means for injecting a supply of additional working fluid into the initial stages of the compressor at speeds below the design speed, may also have means to adjust the pitch of the stator blades of an initial stage of the compressor blading to increase the rotational swirl of the fluid leaving them, at speed below the design speed. Preferably the injection means and the pitch-adjusting means are arranged for simultaneous operation. I

According to yet another feature of the invention the compressor may be provided with adjustable-pitch stator blades preceding the rotor blades of the first stage of compressor blading and the means to supply additional working fluid may be injected into the compressor immediately prior to said adjustable stator blades preferably with to increase the velocity of Working fluid over the radially a tangential or swirl component of velocity whereby the working fluid is given an angle of incidence relative to the adjustable pitch stator blades which is such as to avoid stalling of the stator blades.

Some embodiments of this invention and a theoretical explanation of the effect obtained will now be described with reference to the accompanying drawings in which:

Figure 1 indicates various vectors associated with the first stage blading near the root,

Figure 1a and 1b are velocity triangles for the vectors of Figure 1 at high and low compressor speeds respectively,

Figure 2 indicates various vectors associated with the first stage blading near the tip,

Figures 2a and 2b are velocity triangles for the vec=' rotor blades, and it will be appreciated that the change in whirl velocity Vw, multiplied by the peripheral velocity U and by mass flow, represents a measure of the rate of work done by the root portion of the rotor blade on the working fluid.

The subscript H throughout refers to design or highspeed condition and subscript L to a low-speed condition, i. e. at a speed substantially below the design speed.

It will be appreciated that in the design condition the blades are so arranged that the angles of incidence of the working fluid at entry both to the first stage rotor blades a and to the first stage stator blades 12a have suitable values for the attainment of the desired pressure ratio, the values being such that the angle of incidence at entry is substantially less than the stalling incidence.

When the speed of the compressor is reduced from the design speed to the low speed, the value of the axial velocity VA is diminished relative to the peripheral speed U as indicated by the ratio VAL/ UL as compared with the ratio VAH/UH, and it will be seen that the angle of vector V1L, the resultant velocity of the working fluid relative to the rotor blade at its inlet, is increased relatively to the axial direction as compared with VIH and the blade is working at a higher angle of incidence. In the same way the angle of vector VSL, the resultant velocity of the working fluid relative to the first stage stator blades at inlet thereto, is also increased as compared with Van so that these blades also are working at a higher incidence.

The ratio of the change in whirl velocity Vw to the peripheral speed U and therefore the ratio of the work done by the root section of the blade to the peripheral speed is also slightly increased, as indicated, from Vwrr/ UH to VwL/UL.

Triangles showing corresponding velocities for the tip section of the blade are drawn in Figures 2a and 2b, and it will be appreciated that the ratio of the peripheral speed of this section to the peripheral speed of the blades at the root section will be the same as the ratio of the radius of the tip to the radius of the root.

It is also found that under all conditions when the working fluid is taken in from atmosphere only, the axial velocity Va, at the tip of the blade is less than the axial velocity at the root of the blade, owing to what is known as a centrifugal pressure gradient. The static pressure at the tip section of the blade is greater than that at the root section of the blade owing to centrifugal effect, by an amount given by the expression Since the total head pressure from root to tip is substantially constant at entry to the compressor (from atmosphere) it follows that the resultant velocity of the working fluid at the tip will be less than at the root. The total head pressure from root to tip is also substantially constant at the entry to succeeding stages of the compressor, in normal designs. Thus for a given constant whirl velocity of working fluid at the tip and at the root, the axial velocity of working fluid at the tip will be less than the axial velocity at the root. If, as isusually the case, the whirl component is greater at the tip than at the root, it follows that the axial velocity will be still further reduced at the tip as compared with that at the root.

In this connection it may be pointed out that while the use of adjustable-pitch stator blades to increase the swirl at low rotational speeds in general produces improved operation, in compressors with low hub/tip ratios the adverse effect of the centrifugal pressure gradient may outweight the beneficial effect of increased swirl, thereby producing tip stalling. In this case it may be found desirable to provide the combination of fluid injection in accordance with the present invention and increased swirl, as described hereinafter with reference to Figure 8.

Due to the lower value of the ratio of VA/ U at the tip of the blades, it will be seen from Figure 2 that the triangles are elongated in a horizontal direction as shown in the drawing.

When the axial velocity is reduced, due to the speed of the compressor being reduced from the design condition, from the value represented by VAH to that represented by VAL, the change in whirl velocity is greatly increased relative to the respective peripheral speeds from the value Vwn to the value VWL. The angle of the resultant velocity of the working fluid relative to the rotor blades at inlet is increased from that indicated by line Vm' to that indicated by line V1L, and the angle of incidence with respect to the rotor blade is increased. It will also be seen that the angle of incidence of the first stage stator blade is similarly increased. Owing to the high pitch at the tips of the blades, it has been found that the eflect of a given increase of angle of incidence has a much more adverse effect at the tip than at the root.

It has been found that when the blade angles are designed to give desired conditions at the design speed of the compressor, the angles of incidence at the tip of the blades are increased at the low speed condition to such a value that the blades are operating in a stalled condition and the elficiency of the compressor is seriously affected.

It will be seen that the smaller the ratio of the blade root diameter to the blade tip diameter, (the hub/tip ratio) the more severe this effect will be.

The elfect of increased swirl is further to reduce the axial velocity VAL at the tip of the blade, while increasing the velocity VAL at the root, owing to the etfect of centrifugal pressure gradient. The provision of increased swirl may therefore increase the possibility of tip stalling.

The effect of injecting additional working fluid in accordance with the invention is to increase the effective axial velocity over the tip section of the blade, and thereby to increase the vector V0 so that'its magnitude approaches the magnitude VOL which is in the same ratio VOL'/UL to the magnitude of the peripheral speed UL of the rotor as the ratio in the design condition, represented by VOH/UH. The angle of the flow of the working fluid relative to the rotor blade at inlet thereto is thereby restored to a value approaching that at the design condition, as indicated by Vin, and in a similar manner it will be seen that the angle of the velocity of the working fluid relative to the first stage stator blade is also restored to a value approaching that at the design condition.

It will be appreciated that the injection of the additional working fluid is preferably made at an angle corresponding to that of the vector VoL.

It will be appreciated that the axial velocity of the working fluid under low speed conditions of operation of the compressor may be restored to a value approaching that at the design speed either wholly by the injection of additional working fluid upstream of the blades of which it is desired to improve the efliciency, or partly by the injection, of additional working fluid and partly by the bleeding ofi of working fluid in known manner in the later stages of the compressor. The injection of additional working fluid in accordance with the invention may be combined with the variation of the angular position of the stator blades preceding the rotor blades of which it is desired to increase the efliciency and constructions of compressor in which this is done may also comprise means to bleed ofl working fluid from the later stages of the compressor.

It will be appreciated from consideration of the diagrams that the beneficial elfect is not limited to the stage immediately downstream of the injection of additional working fluid, but the incident angles of the blades of succeeding stages are also improved and their efficiency therefore increased.

a cylinder space 66 on the other side of the piston head. The ram cylinder 31b is provided with fluid supply connection 67 which is connected to the delivery pipe 59 by a branch pipe 93, and bores 68, 69 are taken from the connection 67 to the chambers 66 and 65 respectively. The bore 69 to the chamber 65 on the side of the piston head 63 of larger effective area has fitted in it a flow restrictor 70 the purpose of which will be clear from the following.

The control mechanism 60 is, as will be seen, mounted on the end of the ram cylinder 31b adjacent the side of the piston of larger eifective area. The control mechanism 60 comprises a chamber divided into two compartments 71 and 72 by a flexible diaphragm 73, the compartment 71 nearer the ram cylinder 31b being connected through a union 74 and branch pipe 75 to the suction pipe 52 of the engine fuel supply pump 47 and the compartment 72 being connected through a union 76 and pipe 77 to the chamber 54 of the fuel pump 47 into which pressure fluid is delivered at a pressure proportional to the rotational speed of the compressor. The flexible diaphragm 73 is thus loaded in one direction (in a direction towards the ram cylinder 31b) by a pressure which is a function of the compressor actual rotational speed.

The flexible diaphragm 73 is also arranged to be loaded by a spring 78 which is accommodated within the 1101- low stem of the ram piston 31a and for this purpose the end wall of the ram cylinder 31b is formed with an axially-directed neck 79 having therein a bore fitted with a sleeve 80 containing a sliding push rod 81, one end of which bears on an abutment 73a carried by the diaphragm 73 and the other end of which bears against an abutment member 81b for the spring 78. The other abutment for spring 78 is afforded by the ram piston 31a.

The push rod 81 slides in the sleeve 80 and the push rod 81 is provided with seal-forming lands 81a to prevent leakage of pressure fluid from the space 65 of the ram cylinder into the compartment 71 of the control mechanism 60.

It will be clear that as the ram piston 31a moves in the ram cylinder 31b towards the control mechanism 60, the load on the diaphragm 73 due to the spring 78 will increase and the load will depend upon the position of the piston 31a in the ram cylinder 31b. It will also be clear that this spring load will oppose the fluid pressure load of the diaphragm 73.

The flexible diaphragm 73 is also arranged to be loaded by a secondary spring 82 which acts in the same sense as the spring 78. The secondary spring 82 has one abut-- ment on the member 73a and a second abutment on a shoulder formed within the neck 79.

Movements of the flexible diaphragm 73 are communicated to a second push rod 83 slidably mounted in a bush 84 in the wall diaphragm compartment 72 and movements of the push rod 83 are in turn communicated to a half-ball carrier 85 forming part of a valve mechanism. The valve mechanism comprises a half-ball 86 which cooperates with a bleed port 87 to control a bleed flow through bores 88 from the chamber 65 into a space 89 which is connected through a union 90 and flange 91 with the suction pipe 52 of pump 47. The half-ball carrier 85 and push rod 83 are lightly loaded by a spring 92 towards engagement with the flexible diaphragm 73. i The operation of the ram is as follows. Assuming that the compressor is not rotating, then the half ball 86 of the valve will be closed on its seat around the port 87 and the head 63 of the ram piston 31a will be in abutment against the bush 61 at the end of the ram cylinder 31b remote from the control mechanism 60 where it is held by the main spring 78. i

As the engine starts to rotate pressure in the spaces 65 and 66 on each side of the piston head willstart to build up but since there is no leak through the bores 88' and valve 86, 87, the pressuresin these spaces will remain equal and the piston will remain stationary against its left-hand abutment 61. At the same time the pressure in the pump chamber 54 and thus the fluid pressure acting on the diaphragm 73 will increase due to the increase in the engine rotational speed (and thus the compressor rotational speed).

When the fluid pressure acting on the diaphragm 73 has increased sufliciently to overcome the spring 78 and the secondary spring 82 the half-ball 86 will be lifted oif from its seat around the port 87 and pressure fluid will bleed off from the cylinder space 65 so that pressure in this space will fall relative to the pressure in the space 66. When the pressure within the space 65 has fallen sufficiently to compensate for the difference in eflective areas of the sides of the piston head 63 the fluid pressure load on the piston 31a will overcome the effect of the spring 78 and the piston will start to move within the ram cylinder 31b and will continue to move so long as the engine rotational speed (and thus the compressor rotational speed) increases and the movement will continue until at a selected engine rotational speed, the piston head 63 comes up against the other portion 62. For rotational speeds above the selected rotational speed at which the piston head 63 comes against the bush 62 the control mechanism is ineffective.

In employing the above control arrangement with the compressor illustrated in Figures 3-7, the control mechanism will be adjusted so that the piston head 63 comes up against the stop 62 at the rotational speed at which it is desired to cease additional working fluid injection and the ram piston 31a will be so connected to the part 30 that when the piston head 63 comes up against the stop 62, the rotary valve members 24 will have just reached a position in which the ports 25 are completely out of register with the ports 23.

Where the control mechanism illustrated in Figure 9 is to be used to adjust the angle of incidence of the adjustable nozzle guide vanes 13, the ram 31a will be so connected to the arm 42 (Figure 8) that as the piston head moves from against the stop 61 to the stop 62 the nozzle guide vanes 13 will be rotated about their spanwise axes in a manner gradually to reduce their outlet (and thus to reduce the swirl angle of the working fluid) and the control mechanism 60 of the ram will be arranged so that when the piston head 63 comes up against the stop 62 the inlet guide vanes 13 will be in their normal operating position.

When it is desired to control therotary valve members 24 and adjustable guide vanes 13 simultaneously, two rams such as the rams 31 may be employed and in this case the second ram and its control mechanism may be connected to the engine fuel pump 47 in exactly the same way as the ram illustrated in Figure 9. Suitable connections corresponding to the branch pipes 93, 75, 77 and 91 are indicated in Figure 9 at 93a, 75a, 77a and 91a.

By suitably selecting the rates of the springs 78 and 82 in each ram the rotational speeds at which the rams 31 commence and cease to operate may be made the same or difierent as desired.

When it is desired to control the injection of the additional working fluid in accordance with an operating variable of the compressor which varies in accordance with the speed of the compressor rotor, that is to say, which increases as a function of the speed of the compressor on increase of said speed and decreases as a function of said speed on decrease of said speed, the following arrangements may be adopted.

To control the injection of additional working fluid in accordance with corrected rotational speed which, as is.

well known in the art, is equal to the actual rotational speed divided by the square root of the intake temperature and multiplied by a constant, an additional load may be.

applied to valve 86 by means, for example, of a bellows the interior of which is connected to an element in the 13 rotor blades, means to bleed off air from said annular working fluid duct at a location downstream from said inlet end row of rotor blades comprising at least one annular member affording a portion of the wall of said stator casing at said location and affording a manifold encircling the working fluid duct into which working fluid can flow through apertures in said annular member from the working fluid duct, and from which the working fluid can flow through outlet ports therein, a rotary valve member having ports formed therein to cooperate with said outlet ports, said rotary valve member being rotatable to bring the cooperating ports into and out of register, conduit means interconnecting said outlet ports with said slot, and means to rotate the rotary valve member to bring said cooperating ports into register when the rotational speed of the compressor falls below'a selected rotational speed.

7. A multi-stage axial-flow compressor as claimed in claim 6, wherein said annular member also supports the stator blades of an adjacent row of stator blading and the apertures through which working fluid flows from the working fluid duct into the manifold are formed intermediate the stator blades supported thereby, and wherein said rotary valve member is accommodated within the manifold.

8. A multi-stage axial-flow compressor as claimed in claim 6, wherein the means to rotate the rotary valve member comprises speed-responsive means arranged to respond to the rotational speed of the compressor, and means interconnecting said speed-responsive means and said rotary valve member to bring the ports into register for rotational speeds below a selected rotational speed and to move the cooperating ports out of register for rotational speeds above the selected rotational speed.

9. A multi-stage axial-flow compressor comprising a rotor having mounted thereon a plurality of axially-spaced rows of rotor blades, a stator casing surrounding said rotor and having mounted thereon a plurality of axiallyspaced rows of stator blades to alternate with said rows of rotor blades, said rotor and said stator casing defining an annular working fluid duct having an inlet end and an outlet end, said rotor being adapted to be driven at varying rotational speeds, a circumferential wall part supported from said stator casing upstream of the row of rotor blades adjacent the inlet and having formed therein a slot extending around substantially the whole circumference thereof and directed towards said inlet end row of rotor blades, means to bleed off air from said annular working fluid duct at a location downstream from said in let end row of rotor blades comprising a plurality of annular members each affording a portion of the wall of said stator casing at said location and affording a manifold encircling the working fluid duct into which Working fluid can flow through apertures in said annular member from the working fluid duct, and from which the working fluid can flow through outlet ports therein, a plurality of rotary valve members one for each of said annular members, each rotary valve member having ports formed therein to cooperate with said outlet ports, said rotary valve members being rotatable to bring the cooperating ports into and out of register, conduit means interconnecting said outlet ports with said slot, and means to rotate the rotary valve members simultaneously to bring said cooperating ports into register when the rotational speed of the compressor falls below a selected rotational speed.

10. A multi-stage axial-flow compressor as claimed in claim 9, wherein each said annular member also supports the stator blades of an adjacent row of stator blading and the apertures through which working fluid flows from the working fluid duct into the manifold are formed intermediate the stator blades supported thereby, and wherein each rotary valve member is accommodated in the associated manifold.

11. A multi-stage axial-flow compressor as claimed in claim 9, wherein the means to rotate the rotary valve members comprises speed-responsive means arranged to respond to the rotational speed of the compressor, and means interconnecting said speed-responsive means and said rotary valve members to bring the ports into register for rotational speeds below a selected rotational speed and to move the cooperating ports out of register for rotational speeds above the selected rotational speed.

12. A multi-stage axial-flow compressor comprising a rotor having mounted thereon a plurality of axially-spaced rows of rotor blades, a stator casing surrounding said rotor and having mounted thereon a plurality of axiallyspaced rows of stator blades to alternate with said rows of rotor blades, said rotor and said stator casing defining an annular working fluid duct having an inlet end and an outlet end, said rotor being adapted to be driven at varying rotational speeds, a circumferential wall part supported from said stator casing upstream of the row of rotor blades adjacent the inlet end, and having formed therein a slot extending around substantially the whole circumference thereof and directed towards said inlet end row of rotor blades, porting means in said stator casing and opening to said working fluid duct at a location axially downstream of said inlet end row of rotor blades, conduit means interconnecting said slot with said porting means, said conduit means being afforded by longitudinal channels in the stator casing, valve means arranged to control flow in said conduit means, and means to actuate said valve means to permit a flow of fluid in said conduit means when the rotational speed of the compressor falls below a selected rotational speed.

13. A multi-stage axial-flow compressor comprising a rotor having mounted thereon a plurality of axially-spaced rows of rotor blades; a stator casing having an outer wall, a plurality of longitudinally-extending strengthening ribs internally of said wall, a plurality of axiallyspaced spacer rings integral with said strengthening ribs and a plurality of axially-spaced rows of stator blading supported with their root ends extending between said spacer rings, said rotor being supported within said stator casing with the spacer rings affording shrouds for the rotor blades and the rows of stator blading alternating with the rows of rotor blading; said rotor and said stator casing defining an annular working fluid duct having an inlet end and an outlet end; a circumferential Wall part supported from said stator casing upstream of the row of rotor blades adjacent the inlet end of said casing, said circumferential wall part having formed therein a slot extending around substantially the whole circumference thereof and directed towards said inlet end row of rotor blades, said slot being in communication with the spaces between said longitudinal strengthening ribs adjacent the inlet end of said stator casing; porting means in a spacer ring axially spaced downstream from said inlet end row of rotor blades; valve means arranged to control flow of working fluid through said porting means into the spaces between said longitudinal strengthening ribs; and means to actuate said valve means.

14. A multi-stage axial-flow compressor as claimed in claim 13, wherein said circumferential wall part comprises a portion having at its forward end an inwardly and rearwardly curved surface, an annular vane member coaxial with and rearwardly spaced from said inwardly and rearwardly curved surface, and a portion having an inwardly and rearwardly-curved leading edge rearwardly spaced from said vane member.

15. A multi-stage axial-flow compressor comprising a rotor having mounted thereon a plurality of axially-spaced rows of rotor blades, a stator casing surrounding said rotor and having mounted thereon a plurality of axiallyspaced rows of stator blades to alternate with said rows of rotor blades, said rotor and said stator casing defining an annular working fluid duct having an inlet end and an outlet end, said rotor being adapted to be driven at varying rotational speeds, a circumferential wall part supported from said stator casing upstream of the row of rotor blades adjacent the inlet end and having formed 

